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Archive for February, 2016

Now that I have the Mittag-Leffler universal autocorrelation model for financial time series, it is worth investigating how this can help us in the relationship between gold and mining stocks. In this case an equal-weighted mining stock index returns have a correlation of around 0.2 with daily gold returns. The immediate feature of the mining index is that the autocorrelation is -0.09 while the autocorrelation of gold is 0.1. Our mining companies are PG, PRU, NEM, GOLD, WAC, R, ROG, RPM, between 2013-07-01 and 2016-01-12. Let’s start our analysis by looking at the volatility equalized difference.

We can then look at more refined predictions of volatility using an AR(12) model using stochastic volatility.

Now we can consider this a generic theorem for four space-dimensional manifolds and specialize to four-sphere hypersurfaces with the following special : we take the second fundamental form 3×3 matrix and diagonalize it, setting . Miller’s arguments for unification of electromagnetism and gravitation is unchanged given that the vector potential is given in this special form.

Both the Dirac and Schroedinger equation of quantum mechanics are wave equations with potentials. A natural conjecture is that all such potentials arise as second fundamental form terms from a hypersurface of . The formula for Dirac operator for a hypersurface is well known and presented by Hijazi (hijazi-hypersurface-dirac) for example:

Suppose is a submanifold of and we diagonalize the second fundamental form to obtain three real-valued functions . Now use the eigenvectors on a coframe adapted to , i.e. the last being the dual of the unit normal. If the never vanish, then by the Poincare-Hopf theorem the Euler characteristic is zero for ; this is automatic since is three dimensional. Any orientable closed three-manifold is parallelizable, so this is not a giant restriction. Now consider cohomology . If this is zero, we have a homology sphere which is diffeomorphic to by Poincare Conjecture. In this case, we can look at the Hodge decomposition of 2-forms . There is a unique 3-form such that . Now consider the heat flow starting at . This will lead to a harmonic solution as .

I don’t have a clear answer yet — of course this is a question that led to many failed unified theories both by Weyl and Kaluza-Klein so there is no embarrassment in confusion. So if we take the analogy to Kaluza-Klein, for example the nice paper of Jonah Miller (jonah-miller-kaluza-klein-theorykaluza-elementaryWeyl-Gravitation_and_Electricity), we find that the metric in 5D spacetime is (Greek indices go in his notaion between 0 and 3 with 0 representing time)

What would be the analogue for four-sphere? It would have to be the diagonalized entries of the second fundamental form as .

I have a fairly simple picture of how our actual universe ought to work. I know that the universe has four macroscopic space dimensions and it is embedded in a large radius (radius ) sphere. So then I ask myself, is the electromagnetic tensor really the curvature of a connection? There’s a physical circle normal to every point in the physical universe in my view. A nice theorem tells me that the second fundamental form is globally diagonalizable for a hypersurface of . Then I look at the Gauss and Codazzi equations for the hypersurface

Now these are analogous to the vector potential of electromagnetism. Recall that in the latex curvature picture one looks at vector potentials and considers . In submanifold geometry the second fundamental form terms give us as a modification to the Levi-Civita connection on the hypersurface and formally are identical in form to the electromagnetic potential. From this point of view, a natural candidate for the electromagnetic tensor should be from here but this is standard:

There is thus a natural 2-form that occurs and we want to know whether this could be a good candidate for the electromagnetic 2-form.

by using the product rule and the Codazzi equation by collecting terms by the non-differentiated factor, i.e.

then for example the term with is .

This is still sloppy. However, the comparison is to the gravitational field equations with the electromagnetic stress-energy tensor. This reads

with . I am not sure if this formula has experimental support but it seems strange because it is a function of rather than it’s potential. I wonder if the relationship between gravity and electromagnetism is simpler if one considers the electromagnetic potential directly as a real physical object which would make geometric sense.

A great article on Weyl’s development of electromagnetism is here: varadarajan-weyl-em. He introduced the notion of an electromagnetic field being the curvature of a connection on a -bundle. Formally, one can consider the standard Levi-Civita connection for a hypersurface and write the connection:

^4}_X Y = \nabla^M_X Y + h(X,Y)$

where the second fundamental form term acts just like the electromagnetic potential in the established connection-on-principal bundle approach and yields . In subkmanifold geometry, this is a known issue (see CriticalPointTheory Chapter 2). Thus formally, the principal bundle approach is identical to the normal connection approach where the vector potential is simply the shape operator.